A192896 -id:A192896 - cp's OEIS frontend

A191749 Numbers not the sum of a smaller number and its prime factors (with multiplicity).

1, 2, 3, 5, 7, 9, 12, 13, 16, 18, 20, 21, 25, 27, 28, 30, 32, 37, 43, 44, 45, 48, 49, 50, 52, 57, 60, 61, 64, 66, 67, 68, 70, 73, 75, 77, 78, 80, 81, 85, 87, 90, 91, 92, 97, 100, 101, 102, 104, 108, 110, 112, 115, 117, 126, 129, 130, 132, 133, 135, 137, 139, 144, 145

Offset: 1

Alonso del Arte, Jul 13 2011

nonn

If a number is not squarefree, then its repeated prime factors are added as many times as the exponent indicates (e.g., the sum of prime factors of 8 is 6 since 8 = 2 * 2 * 2 and 2 + 2 + 2 = 6).

No even semiprime (A100484) can be in this sequence, since, if nothing else, it is the sum of a prime number and that prime number's only prime factor (itself).

			3 is in the sequence since neither 1 + sopfr(1) nor 2 + sopfr(2) add up to 3 (instead these equal 2 and 4 respectively).
Because 2 + sopfr(2) = 4, the number 4 is not in this sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A096461, A192896 (only a(1) of those sequences can be in this sequence). Cf. also A001414. Analogous to A005114.

pfAddSeq[start_, max_] := NestWhileList[# + Plus@@Times@@@FactorInteger@# &, start, # < max &]; Complement[Range[200], Flatten[Table[Drop[pfAddSeq[n, 200], 1], {n, 2, 200}]]] (* corrected by Amiram Eldar, Aug 14 2025 *)

upto(n) = {
	v = vector(n);
	for(i = 2, n,
		c = i + sopfr(i);
		if(c <= n,
			v[c] = 1));
	select(x -> x == 0, v, 1)}
sopfr(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i,1] * f[i,2])} \\ David A. Corneth, Aug 14 2025

2 inserted by and more terms from David A. Corneth, Aug 14 2025

cp's OEIS Frontend

A191749 Numbers not the sum of a smaller number and its prime factors (with multiplicity).

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